On maximal functions in Orlicz spaces

Abstract

Let Φ ( t ) \Phi (t) and Ψ ( t ) \Psi (t) be the functions having the representations Φ ( t ) = ∫ 0 t a ( s ) d s \Phi (t)=\int _{0}^{t} a(s)ds and Ψ ( t ) = ∫ 0 t b ( s ) d s \Psi (t)=\int _{0}^{t} b(s)ds , where a ( s ) a(s) is a positive continuous function such that ∫ 1 ∞ a ( s ) s d s = + ∞ \int _{1}^{\infty }\frac {a(s)}{s}ds=+\infty and b ( s ) b(s) is quasi-increasing. Then the maximal function M f Mf is a function in Orlicz space L Φ L^{\Phi } for all f ∈ L Ψ f\in L^{\Psi } if and only if there exists a positive constant c 1 c_{1} such that ∫ 1 s a ( t ) t d t ≤ c 1 b ( c 1 s ) \int _{1}^{s} \frac {a(t)}{t}dt\leq c_{1}b(c_{1}s) for all s ≥ 1 s\geq 1 .